Del Pezzo surfaces with $$\frac{1}{3}(1,1)$$ 1 3 ( 1 , 1 ) points

Abstract

We classify non-smooth del Pezzo surfaces with 1 3 (1, 1) points in 29 qG-deformation families grouped into six unprojection cascades (this overlaps with work of Fujita and Yasutake [14]), we tabulate their biregular invariants, we give good model constructions for surfaces in all families as degeneracy loci in rep quotient varieties, and we prove that precisely 26 families admit qG-degenerations to toric surfaces. This work is part of a program to study mirror symmetry for orbifold del Pezzo surfaces [2].In this paper we:(I) Classify non-smooth del Pezzo surfaces with 1 3 (1, 1) points in precisely 29 qG-deformation families. We further structure the classification into six unprojection cascades, determine their biregular invariants and their directed MMP together with a distinguished configuration of negative curves on the minimal resolution. This overlaps with work of Fujita and Yasutake [14].(II) Construct good models for surfaces in all families as degeneracy loci in rep quotient varieties. In all but two cases, the rep quotient variety is a simplicial toric variety.(III) Prove that precisely 26 of the 29 families admit a qG-degeneration to a toric surface.The classification is summarised in table 1 and table 2, which also plot invariants and provide good model constructions of surfaces in all families.This work is part of a program to understand mirror symmetry for orbifold del Pezzo surfaces [2,16,26,31,27,10] and it is aimed specifically at giving evidence for the conjectures made in [2].The rest of the introduction is organised as follows: in § 1.1 we give precise statements of our main results; in § 1.2 we say a few words about the context of [2]; in § 1.3 we outline the structure of the paper. Our results1.1.1 The classification and its cascade structuren acts linearly on C 2 with weights a, b ∈ 1 n Z /Z. We always assume no stabilisers in codimension 0, 1, that is, hcf(a, n) = hcf(b, n) = 1. A del Pezzo surface is a surface X with cyclic quotient singularities and −K X ample. The Fano index of X is the largest positive integer f > 0 such that −K X = f A in the Class group Cl X.Remark 2. In this paper we view a del Pezzo surface X with quotient singularities as a variety. Such a surface is in a natural way a smooth orbifold (or DM stack), but we mostly ignore this structure. Thus for us Cl X is the Class group of Weil divisors on X modulo linear equivalence. In particular, although K X is a Cartier divisor on the orbifold, we think of it as a Q-Cartier divisor on the underlying variety (the coarse moduli space of the orbifold) and then to say that it is ample is to say that a positive integer multiple is Cartier and ample. See [2] for a discussion of qG-deformations of del Pezzo surfaces with cyclic quotient singularities. In particular, it is explained there that the singularity 1 3 (1, 1) is qG-rigid and the degree d = K 2 is locally constant in qG-families.We classify qG-deformation families of del Pezzo surfaces with k ≥ 1 1 3 (1, 1) points. It follows for example from the proof of [2, Lemma 6] tha...

Laurent inversion

“…Indeed, regardless of its context, Laurent inversion gives a powerful new method for constructing algebraic varieties. We illustrate this in §10 below, where we exhibit explicit models for del Pezzo surfaces with 1/3(1, 1) singularities that played an essential role in the Corti-Heuberger classification [18], and which are hard to construct using more traditional methods.…”

Restrictions on the singularity content of a Fano polygon

“…Recent results support this conjecture [4,7,9], and understanding the possible values taken by the singularity content is an important open question. As a first step towards addressing this question, the two main results of this paper are:…”

Del Pezzo surfaces with a single $1/k(1,1)$ singularity

“…The roots of R 8 3 are tabulated below, and recall that we are free to permute the b i and reverse signs to generate roots from the ones listed in this table. The proof of Theorem 1.1 is based on the directed MMP and has an identical structure to the classification of del Pezzo surfaces with 1 3 (1, 1) singularities in [12], although our current task is made considerably simpler by the assumption there is a single 1 k (1, 1) singularity. Definition 6.1.…”